General Lower Bounds for the Running Time of Evolutionary Algorithms

2010 ◽  
pp. 124-133 ◽  
Author(s):  
Dirk Sudholt
Keyword(s):  
Evolutionary Algorithms ◽  
Lower Bounds ◽  
Running Time

scholarly journals A General Approach to Running Time Analysis of Multi-objective Evolutionary Algorithms

2018 ◽  
Author(s):  
Chao Bian ◽  
Chao Qian ◽  
Ke Tang
Keyword(s):  
Evolutionary Algorithms ◽  
Optimization Problems ◽  
Theoretical Aspect ◽  
Combinatorial Problems ◽  
Time Analysis ◽  
Running Time ◽  
Asymptotic Bounds ◽  
Multi Objective ◽  
Running Time Analysis ◽  
Empirical Performance

Evolutionary algorithms (EAs) have been widely applied to solve multi-objective optimization problems. In contrast to great practical successes, their theoretical foundations are much less developed, even for the essential theoretical aspect, i.e., running time analysis. In this paper, we propose a general approach to estimating upper bounds on the expected running time of multi-objective EAs (MOEAs), and then apply it to diverse situations, including bi-objective and many-objective optimization as well as exact and approximate analysis. For some known asymptotic bounds, our analysis not only provides their leading constants, but also improves them asymptotically. Moreover, our results provide some theoretical justification for the good empirical performance of MOEAs in solving multi-objective combinatorial problems.

scholarly journals Clique Is Hard on Average for Regular Resolution

2021 ◽  
Vol 68 (4) ◽  
pp. 1-26
Author(s):  
Albert Atserias ◽  
Ilario Bonacina ◽  
Susanna F. De Rezende ◽  
Massimo Lauria ◽  
Jakob Nordström ◽  
...  
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
State Of The Art ◽  
Edge Density ◽  
Multiplicative Constant ◽  
Running Time ◽  
Maximum Cliques

We prove that for k ≪ 4√ n regular resolution requires length n Ω( k ) to establish that an Erdős–Rényi graph with appropriately chosen edge density does not contain a k -clique. This lower bound is optimal up to the multiplicative constant in the exponent and also implies unconditional n Ω( k ) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.

scholarly journals On The Independence Number Of Some Strong Products Of Cycle-Powers

2015 ◽  
Vol 40 (2) ◽  
pp. 133-141 ◽  
Author(s):  
Marcin Jurkiewicz ◽  
Marek Kubale ◽  
Krzysztof Ocetkiewicz
Keyword(s):  
Lower Bounds ◽  
Independence Number ◽  
Computational Results ◽  
Shannon Capacity ◽  
Noisy Channels ◽  
Running Time ◽  
The Third ◽  
Independence Numbers

Abstract In the paper we give some theoretical and computational results on the third strong power of cycle-powers, for example, we have found the independence numbers α((C102)√3) = 30 and α((C144)√3) = 14. A number of optimizations have been introduced to improve the running time of our exhaustive algorithm used to establish the independence number of the third strong power of cycle-powers. Moreover, our results establish new exact values and/or lower bounds on the Shannon capacity of noisy channels.

scholarly journals Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift

2020 ◽  
pp. 1-25
Author(s):  
Benjamin Doerr
Keyword(s):  
Evolutionary Algorithms ◽  
Lower Bounds ◽  
Population Based ◽  
Mutation Rates ◽  
Reproduction Rate ◽  
Random Mutation ◽  
Search Spaces ◽  
Discrete Search ◽  
Negative Drift ◽  
Special Case

A decent number of lower bounds for non-elitist population-based evolutionary algorithms has been shown by now. Most of them are technically demanding due to the (hard to avoid) use of negative drift theorems — general results which translate an expected movement away from the target into a high hitting time. We propose a simple negative drift theorem for multiplicative drift scenarios and show that it can simplify existing analyses. We discuss in more detail Lehre's (PPSN 2010) negative drift in populations method, one of the most general tools to prove lower bounds on the runtime of non-elitist mutation-based evolutionary algorithms for discrete search spaces. Together with other arguments, we obtain an alternative and simpler proof of this result, which also strengthens and simplifies this method. In particular, now only three of the five technical conditions of the previous result have to be verified. The lower bounds we obtain are explicit instead of only asymptotic. This allows to compute concrete lower bounds for concrete algorithms, but also enables us to show that super-polynomial runtimes appear already when the reproduction rate is only a [Formula: see text] factor below the threshold. For the special case of algorithms using standard bit mutation with a random mutation rate (called uniform mixing in the language of hyper-heuristics), we prove the result stated by Dang and Lehre (PPSN 2016) and extend it to mutation rates other than [Formula: see text], which includes the heavytailed mutation operator proposed by Doerr, Le, Makhmara, and Nguyen (GECCO 2017). We finally use our method and a novel domination argument to show an exponential lower bound for the runtime of the mutation-only simple genetic algorithm on ONEMAX for arbitrary population size.

Exponential Lower Bounds for the Running Time of DPLL Algorithms on Satisfiable Formulas

2006 ◽  
Vol 35 (1-3) ◽  
pp. 51-72 ◽  
Author(s):  
Michael Alekhnovich ◽  
Edward A. Hirsch ◽  
Dmitry Itsykson
Keyword(s):  
Lower Bounds ◽  
Running Time
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